If I am not in a different country (not my country) and want to get married in that country and do not have my birth certificate with me can I still get married with only my passport?

It depends on
1. what county you are from
2. what county you are in
3. the rules of that county you are in, and if in the US, the rules of that state within the US.

4. Also be aware that in many places just getting married, unless you have gotten the correct VISA's it does not mean you have any rights to stay in the county

If the person is canadian and does not want to go back to canada just to get the birth certificate can they just use the passport to get married?